具有虚乘法的阿贝尔三维簇的剩余表示的计算
Computing residual representations of abelian threefolds with imaginary multiplication
- University of Wisconsin-Madison（威斯康星大学麦迪逊分校）
- Max Planck Institute for Mathematics（马克斯·普朗克数学研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究虚二次域上具有虚乘法的阿贝尔三维簇的剩余伽罗瓦表示，提出大像判据与高效算法，应用于大量曲线并实现酉半相似群为伽罗瓦群，同时发现新的 Picard 曲线族。
AI中文摘要：
设 $M$ 为虚二次域。设 $A$ 为定义在 $M$ 上的极化阿贝尔三维簇，其几何自同态代数同构于 $M$。我们研究附着于 $A$ 的剩余伽罗瓦表示。特别地，我们描述了自同态域以及由自同态对剩余表示施加的自然限制。然后，我们设计了剩余伽罗瓦表示像为大的判据，并提供了适用于大规模计算的高效算法。随后，我们将此算法应用于多个族中的数百万条曲线，这些曲线的雅可比簇是具有虚乘法的阿贝尔三维簇，并应用于 Sutherland 的 $7$-光滑 Picard 曲线数据集。我们给出了一个 Picard 曲线的显式例子，该曲线似乎具有定义在 $\mathbb{Q}(\zeta_3)$ 上的 $13$ 次同源。在我们计算范围内，所有其他具有自同态代数 $M$ 的曲线的雅可比簇，对于任何素数 $\ell>7$，其模 $\ell$ 像尽可能大。这使我们能够对所有 $\ell \not\equiv 1, 25, 121 \pmod{168}$，将酉半相似群 $\Gamma\mathrm{U}_3(\ell)$ 实现为 $\mathbb{Q}$ 上的伽罗瓦群。我们的算法还引导我们发现了几类有趣的 Picard 曲线有理族，其一般成员似乎具有维数为 $6$ 的自同态代数，且该代数不是 CM 域。这些曲线表示 Picard 模曲面上的曲线，似乎是 Shimura 曲线。其中一些似乎参数化具有四元数乘法的非主极化阿贝尔曲面。
英文摘要：
Let $M$ be an imaginary quadratic field. Let $A$ be a polarised abelian threefold defined over $M$ with geometric endomorphism algebra isomorphic to $M$. We study residual Galois representations attached to $A$. In particular, we describe the endomorphism field and the natural restrictions placed on the residual representations by their endomorphisms. We then devise criteria for the image of residual Galois representations to be large, and provide efficient algorithms suitable for large scale calculations. Subsequently, we apply this algorithm to millions of curves in several families, whose Jacobians are abelian threefolds with imaginary multiplication, and to Sutherland's dataset of $7$-smooth Picard curves. We produce an explicit example of a Picard curve which appears to have an isogeny of degree $13$ defined over $\mathbb{Q}(ζ_3)$. The Jacobians of all other curves, in the range of our computation with endomorphism algebra $M$, have mod-$\ell$ image as large as possible for any prime $\ell>7$. This allows us to realise the group $Γ\mathrm{U}_3(\ell)$ of unitary semisimilitudes as a Galois group over $\mathbb{Q}$ for all $\ell \not\equiv 1, 25, 121 \pmod{168}$. Our algorithms also led us to the discovery of several interesting rational families of Picard curves whose generic members appear to have endomorphism algebra of dimension $6$ which is not a CM field. These represent curves on the Picard modular surface and appear to be Shimura curves. Some of them appear to parametrise non-principally polarised abelian surfaces with quaternionic multiplication.
